If what you are saying is correct, than what you are describing is, indeed, perfect pitch. Congratulations! If you can consistently identify songs that begin with any particular note, then you could hypothetically memorize the keys of 12 songs and then be able to name any note that you hear.
To be perfectly clear: begin able to name the notes by letter is merely a way to prove that you have perfect pitch. You can have perfect pitch without being able to name the notes. An individual with no musical training (and who therefore is unaware of note names broadly) can still have perfect pitch. It would just be harder to verify.
Also be aware that this is not a strict binary. You don't either have perfect pitch or not have perfect pitch. You can have weaker or stronger perfect pitch, and you can (to some degree) train to improve what you have. Folks with extremely strong perfect pitch can sometimes identify how many cents off an out-of-tune note is from the standard, and can name notes even when there are distracting elements present (such as other notes playing).
By contrast, if you had to mentally refer back to one of the 4 violin notes, and could only identify notes that were played on their own (with no harmonic context), then this would be a weaker skill. However, as mentioned, you could still improve it with training.